Are these two definitions of the state-action value function equivalent?
These two definitions are not exactly the same, even though they have a very similar formulation. David Silver's notation is probably an abuse of notation.

The first difference between those two definitions is that, in the case of David Silver's slides, the policy is parametrized by $\theta$ (i.e. the policy could be represented e.g. by a neural network), while, in the first case, the policy is not parametrized, which is a more general way of writing the definition of the state-action value function, because you do not assume how you will represent the policy. 

The second apparent difference (as you have noticed) is that, in the first case, the expectation is taken with respect to transition probability function, while, in the second case, it seems to be taken with respect to the parametrized policy. In the second case, you do not probably know the transition probability function, so you will compute your state-action value function based on your current best estimate of the policy.

Now, as Brale pointed out in the comments (please, read the comments under this answer!), the definition of the state-action value function is always the same, even if you don't know the MDP (because that's a definition and it doesn't matter if you know the transition probability or not). However, in the case of David Silver's slides, by that notation, I think he means he will compute the state-action value function based on the current estimate of the policy.